#72226070

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Splittings

Sylow

C₃ ⋊ #141837

Subgroups

	subgroup	G/H
center	C₂²	#71764
commutator	C₃×C₄×C₈	C₂²×C₄
Frattini	C₂×C₄×C₈	C₂²×D₃
Fitting, prime radical	C₃×#19603	C₂
socle	C₂²×C₃	#1957
cosocle	C₂×C₃×C₄×C₈	C₂³
hypercenter	#19603	D₃
hypocenter	C₃	#141837
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #19603

734 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3)(4,6,15,19,8,14,7,16)(5,10,9,17,11,18,12,13);(2,3)(5,9,11,12)(6,14)(10,13,18,17)(16,19)(20,21,22,27,24,29,32,34,25,30,33,26,23,28,31,35);(4,5)(6,13)(7,9)(8,11)(10,16)(12,15)(14,17)(18,19)(21,26)(22,31)(23,24)(27,30)(28,35)(29,34)(32,33)

Polycyclic presentation:

2^9,3|b^2=d,c^2=efg,d^2=h,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cefhi,c^b=cgh,d^a=dg,e^a=eg,e^b=ei,f^a=fhi,f^b=fi,g^a=gi,h^a=hi,j^b=j^2

Conjugacy classes & character table

144 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226076, #72226088, #72226202
maximal normal subgroup of	#72226076, #72226088, #72226202

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5*F.6*F.7);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);